Numerical Modeling of the Interaction of Solitary Waves and Submerged Breakwaters with Sharp Vertical Edges Using One-Dimensional Beji & Nadaoka Extended Boussinesq Equations

Using one-dimensional Beji & Nadaoka extended Boussinesq equation, a numerical study of solitary waves over submerged breakwaters has been conducted. Two different obstacles of rectangular as well as circular geometries over the seabed inside a channel have been considered in view of solitary waves passing by. Since these bars possess sharp vertical edges, they cannot directly be modeled by Boussinesq equations. Thus, sharply sloped lines over a short span have replaced the vertical sides, and the interactions of waves including reflection, transmission, and dispersion over the seabed with circular and rectangular shapes during the propagation have been investigated. In this numerical simulation, finite element scheme has been used for spatial discretization. Linear elements along with linear interpolation functions have been utilized for velocity components and the water surface elevation. For time integration, a fourth-order Adams-Bashforth-Moulton predictor-corrector method has been applied. Results indicate that neglecting the vertical edges and ignoring the vortex shedding would have minimal effect on the propagating waves and reflected waves with weak nonlinearity. 1. Introduction Boussinesq type equations are among the most practical mathematical models used in offshore engineering. These equations include nonlinear terms as well as dispersion terms. Thus, they are one of the most robust tools for hydrodynamic study of nearshore waves. During the years from 1871 to 1872, Boussinesq introduced these equations by adding dispersion effects to the shallow water equations originally known as Saint Venant. These equations have a hyperbolic structure with derivatives of high order in order to numerically model the dispersion-based physics. Peregrine [1] introduced what is known as the basic type of Boussinesq equations. Using Boussinesq equations for inviscid fluids, continuity equation with integral representation and applying respective boundary conditions, the basic Peregrine-Boussinesq equations for long waves over variable seabeds can be derived. Many efforts have been made for the development of Boussinesq equations. These efforts have been made with the aim of enhancing dispersion (ratio of water depth to wave length) characteristics of the equations in order to preserve their validity for deep waters applications. As one of the first attempts, Witting [2] used the momentum equations on the integrated depth in one dimension so he could present practical equations based on the velocity terms that were defined on the free surface. In the governing